Limit Distribution of the OLS Estimator

Normality as an Asymptotic Approximation

Vladislav Morozov

Introduction

Lecture Info

Learning Outcomes

This lecture is about a


By the end, you should be able to

  • R

Textbook References

  • Refresher on probability:
    • Your favorite probability textbook
    • Section B in Wooldridge (2020)
  • Asymptotic theory for the OLS estimator
    • E4 in Wooldridge (2020)
    • Or chapter 7 in Hansen (2022)

Motivation

Goal

inference

  • Quantify uncertaintY: confidence intervals
  • Test something

Reminder:

normality

Issue with Normality

But normality

Example: positive outcomes and positive regressors: how can \(U_{it}\) be normal?

Issue: Unknown Distribution

Normality was useful?

If we knew some other distribution, would be nice

But we cannot

Options

  • Nonasymptotic/finite-sample analysis based on “high-probability bounds” — originally more popular in high-dimensional Wainwright (2019)
  • Large-sample “approximations” using tools like the central limit theorem(s) — topic of this lecture

?

Asymptotic Normality

Model

Model

Hansen, Bruce. 2022. Econometrics. Princeton_University_Press.
Wainwright, Martin J. 2019. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge University Press. https://doi.org/10.1017/9781108627771.
Wooldridge, Jeffrey M. 2020. Introductory Econometrics: A Modern Approach. Seventh edition. Boston, MA: Cengage.